The document discusses probability concepts using mathematical equations and inequalities. It explores different probability distributions and scenarios, deriving optimal probability values that satisfy given constraints. For example, it finds the probability μ that maximizes an expected value given the constraints μ > 3/5 and μ < 3/5, determining the optimal probability is 3/5. It also examines cases where the optimal probability is 1/2.

2. “ ”
— —

8.
=
p(x|b)
p(u|b)
=
1/2
1
= 1/2

9.
=
p(x|b)
p(u|b)
=
0
1/2
= 0

10.
=
p(x|b)
p(u|b)
=
1/2
1/2
= 1

11.
=
p(x|b)
p(u|b)
=
1/2 * 2/3
1/2 * 2/3 + 1/2 * 1/2
=
4
7

15.
2 * μ + (−1) * (1 − μ) = 3μ − 1
0 * μ + 2 * (1 − μ) = − 2μ + 2
3μ − 1 > − 2μ + 2
μ >
3
5
μ <
3
5
μ =
3
5

18.
2μ1 + (−1)μ2 + 2μ3
3μ1 + 3μ2 + 0μ3

20.
μ >
3
5
μ <
3
5
μ =
3
5
μ < 3/5

21.
μ >
3
5
μ <
3
5
μ =
3
5
μ > 3/5

22.
μ >
3
5
μ <
3
5
μ =
3
5
μ < 3/5

23.
μ >
3
5
μ <
3
5
μ =
3
5

27. 2μ + 0(1 − μ) = 2μ
0μ + 2(1 − μ) = 2 − 2μ

28. 2μ + 0(1 − μ) = 2μ
0μ + 2(1 − μ) = 2 − 2μ
μ >
1
2
μ <
1
2
μ =
1
2

29. 2μ + 0(1 − μ) = 2μ
0μ + 2(1 − μ) = 2 − 2μ
μ >
1
2
μ <
1
2
μ =
1
2

30. 2μ + 0(1 − μ) = 2μ
0μ + 2(1 − μ) = 2 − 2μ
μ >
1
2
μ <
1
2
μ =
1
2

31. 2μ + 0(1 − μ) = 2μ
0μ + 2(1 − μ) = 2 − 2μ
μ >
1
2
μ <
1
2
μ =
1
2

33. A B

38. See you next time